Torsion anomalous points and families of elliptic curves
This paper proves that there are at most finitely many complex \(\lambda \not= 0, 1\) such that the two points \(P_\lambda = (2, \sqrt{2(2-\lambda)})\) and \(Q_\lambda = (3, \sqrt{6(3-\lambda)})\) on the Legendre elliptic curve \(E_\lambda : Y^2 = X(X-1)(X-\lambda)\) both have finite order. The proof uses the following two results deduced from \textit{S. David} [J. Number Theory 64, No. 1, 104--129 (1997; Zbl 0873.11035)], and \textit{J. H. Silverman} [J. Reine Angew. Math. 342, 197--211 (1983; Zbl 0505.14035)], respectively, together with estimates on the number of rational points on subanalytic sets \textit{J. Pila} [Q. J. Math. 55, No. 2, 207--223 (2004; Zbl 1111.32004)]. Results of the latter type have been recently used to give new proofs of the Manin-Mumford conjecture, as well as cases of the Andre-Oort conjecture which are unconditional. I. There is an absolute constant \(c\) such that if for some \(\lambda \not= 0, 1\), the point \(P_\lambda\) or \(Q_\lambda\) has finite order \(n\), then \(\lambda\) is algebraic and \(n \leq c d(\lambda)^2 (1 + h(\lambda))\), where \(d = d(\lambda)\) is the degree of \(\lambda\) over \(\mathbb{Q}\), and \(h(\lambda)\) is the standard absolute Weil height of \(\lambda\). II. There is an absolute constant \(c\) such that if for some \(\lambda \not= 0, 1\), the point \(P_\lambda\) or \(Q_\lambda\) has finite order, then \(h(\lambda) \leq c\). The proof proceeds by showing that we can take \(d \leq c\) if both \(P_\lambda\) and \(Q_\lambda\) have finite order. The Northcott property then implies finiteness for the number of \(\lambda\) such that both \(P_\lambda\) and \(Q_\lambda\) have finite order. The inequality \(d \leq c\) is established in the simplest case by using a real analytic map \(\Theta : D \rightarrow \mathbb{R}^4\) where \(D\) is a closed disc in \(\mathbb{R}^2\) containing at least \(\frac{d}{2}\) conjugates of \(\lambda\) and applying the estimates for the number of rational points in \(\Theta(D)\) of the form \(\frac{1}{n} \mathbb{Z}^4\).
scientific article; zbMATH DE number 5277917
scientific article